Hello -- Three related questions, going from most specific to most general : 1 ) Consider the stream processing arrow which computes a running sum, with two implementations : first using generic ArrowCircuits (rSum); second using Automaton (rSum2) : module Foo where import Control.Arrow import Control.Arrow.Operations import Control.Arrow.Transformer import Control.Arrow.Transformer.All rSum :: ArrowCircuit a => a Int Int rSum = proc x -> do rec out <- delay 0 -< out + x returnA -< out rSum2 = Automaton (f 0) where f s n = let s' = s + n in (s', Automaton (f s')) runAuto _ [] = [] runAuto (Automaton f) (x:xs) = let (y, a) = f x in y : runAuto a xs take 10 $ runAuto rSum [1..] [0,1,3,6,10,15,21,28,36,45] take 10 $ runAuto rSum2 [1..] [1,3,6,10,15,21,28,36,45,55] Note that the circuit version starts with the initial value zero. Is there a way to write rSum2 in the general ArrowCircuit form, or using ArrowLoop? 2) Are the ArrowLoop instances for (->), Kleisli Identity, and Kleisli ((->) r) all morally equivalent? (e.g., up to tagging and untagging?) 3) One can define fix in terms of trace and trace in terms of fix. trace f x = fst $ fix (\(m, z) -> f (x, z)) fix f = trace (\(x, y) -> (f y, f y)) undefined Does this mean we can translate arbitrary recursive functions into ArrowLoop equivalents? Best regards, Ben